In this paper, we consider a bootstrap approximation to the distribution of the least squares estimator [~ of the autoregressive parameter [:~ in a first-order autoregressive process which may or may not have a stationary solution. Our bootstrap procedure is a modification of the standard bootstrap and employs a data based shrinkage towards the critical values [t = Β±1.
The error in estimation of the sampling distribution of [i by the above procedure converges to zero as the sample size grows, irrespective of the value of the autoregressive parameter. This result is in sharp contrast with the behavior of the standard bootstrap tbr which a random limiting distribution emerges at the critical values [;~ -i14 resulting in a failure of tile bootstrap procedure. The asymptotic validity of our procedure enables us, inter alia, to construct a confidence interval for [;t which will have the correct coverage probability (asymptotically) under any real [1.
The theoretical validity result for the modified bootstrap is supported by appropriate sinmlations. Finally, we also indicate an extension of the proposed modified bootstrap method to the pth order autoregressive processes.